Welcome aboard, knoggin!
The engine "unit number" is located on the front of the block, on the RH (passenger) side.
There should be the two-letter alpha code (YS indicates a base 400/350HP with automatic trans) and an engine serial number. The last 4 digits of the engine serial number should match the "Engine Unit Number" on the PHS copy of the Car Billing History form. The Billing History will probably have an entry like "YSXXXX", where the XXXX should be the last 4 digits of the engine serial number.
Also, you should find a partial VIN stamped on the front of the block, below the area where the engine code and serial number are located. This will be a vertically oriented stamp next to the front timing cover that should match the last 6 digits of the VIN. This will be just above the oil pan.
A "YS" car would have a TH400 automatic trans (PX code on the trans tag).
You can also confirm rear axle ID and date codes; I find this site very helpful:
Wallace Racing-Drag Racing Pontiac powered Firebirds,Trans Ams and Dragsters
Scroll down and use the various "Code Search" links on the left.
Also verify the information on the Data Plate at the top of the cowl, near the driver's hood hinge. The model will be 24237(hardtop) or 24267(convertible).
Paint codes (upper/lower), interior style and color and build date/plant info is all on the data plate.
Lots more info here: Pontiac GTO - The Ultimate Pontiac GTO Picture Site
Go to the data/specifications page for 1968; lots of helpful details on plant codes, color codes, option codes, interior codes and build dates.
Good luck and happy hunting!